Problem with moving vertices in shape key edit mode
When in Edit mode, there is an Options tab in the Tool box (T).  In the tab you had X mirror and topology mirror turned on.  If you untick X mirror, the two vertices stop moving together.

(Also, you ought to Apply Rotation, (Control A/ Rotation) so that local axes agree with global axes, not that that was the cause of the problem. It can cause rigging problems if it's not applied. Also you should Apply scale (Control A/Scale) because if you ever animate a child object rotating around the bear it will inherit the Scale as it rotates, and become bigger on one axis, then shrink again as it goes further around)



Note that if you apply Rotation and Scale, but don't untick the X mirror box, the two vertices now move in a logical, mirrored direction at least. So your local rotation and scale values were interfering with the proper function of the X mirror tool.